SOLUTION: Please help me solve this equation: You commute 56 miles one way to work. The trip to work takes 10 minutes longer than the trip to home. Your average speed on the trip home is

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Question 173044: Please help me solve this equation:
You commute 56 miles one way to work. The trip to work takes 10 minutes longer than the trip to home. Your average speed on the trip home is 8 miles per hour faster. What is your average speed on the trip home?

thank you!!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Note: 10 minutes = 1/6 of an hour


Since the distance of the commute is 56 miles, this means that d=56



Going to Work:


Since "The trip to work takes 10 minutes longer than the trip to home", this means that the time is t%2B10 (in minutes) or t%2B1%2F6 (in hours)


We'll use the expression t%2B1%2F6

d=rt Start with the distance rate time formula


56=r%28t%2B1%2F6%29 Plug in d=56 and replace "t" with t%2B1%2F6


56=r%28%286t%2B1%29%2F6%29 Combine t and 1%2F6 to get %286t%2B1%29%2F6


56%2F%28%286t%2B1%29%2F6%29=r Divide both sides by %286t%2B1%29%2F6 to isolate "r"


56%286%2F%286t%2B1%29%29=r Multiply 56 by the reciprocal of %286t%2B1%29%2F6.


336%2F%286t%2B1%29=r Multiply 56 and 6 to get 336


So after isolating "r", we get r=336%2F%286t%2B1%29

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Coming Home:


Now because the "average speed on the trip home is 8 miles per hour faster", this means that the new rate is r%2B8



d=rt Go back to the distance rate time formula


56=%28r%2B8%29t Plug in d=56 (the commute distance hasn't changed) and replace "r" with r%2B8 (since the speed is 8 mph greater)



56=%28336%2F%286t%2B1%29%2B8%29t Plug in r=336%2F%286t%2B1%29


56=%28336%2F%286t%2B1%29%29t%2B8t Distribute


56%286t%2B1%29=336t%2B8t%286t%2B1%29 Multiply EVERY term by the LCD 6t%2B1 to clear the fractions.


336t%2B56=336t%2B48t%5E2%2B8t Distribute


0=336t%2B48t%5E2%2B8t-336t-56 Subtract 336t from both sides. Subtract 56 from both sides.


0=48t%5E2%2B8t-56 Combine like terms.


Notice we have a quadratic equation in the form of at%5E2%2Bbt%2Bc where a=48, b=8, and c=-56


Let's use the quadratic formula to solve for t


t+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


t+=+%28-%288%29+%2B-+sqrt%28+%288%29%5E2-4%2848%29%28-56%29+%29%29%2F%282%2848%29%29 Plug in a=48, b=8, and c=-56


t+=+%28-8+%2B-+sqrt%28+64-4%2848%29%28-56%29+%29%29%2F%282%2848%29%29 Square 8 to get 64.


t+=+%28-8+%2B-+sqrt%28+64--10752+%29%29%2F%282%2848%29%29 Multiply 4%2848%29%28-56%29 to get -10752


t+=+%28-8+%2B-+sqrt%28+64%2B10752+%29%29%2F%282%2848%29%29 Rewrite sqrt%2864--10752%29 as sqrt%2864%2B10752%29


t+=+%28-8+%2B-+sqrt%28+10816+%29%29%2F%282%2848%29%29 Add 64 to 10752 to get 10816


t+=+%28-8+%2B-+sqrt%28+10816+%29%29%2F%2896%29 Multiply 2 and 48 to get 96.


t+=+%28-8+%2B-+104%29%2F%2896%29 Take the square root of 10816 to get 104.


t+=+%28-8+%2B+104%29%2F%2896%29 or t+=+%28-8+-+104%29%2F%2896%29 Break up the expression.


t+=+%2896%29%2F%2896%29 or t+=++%28-112%29%2F%2896%29 Combine like terms.


t+=+1 or t+=+-7%2F6 Simplify.


So the possible time values (coming home) are t+=+1 or t+=+-7%2F6


However a negative time does NOT make sense. So the only answer is t+=+1


So the time to commute home is 1 hour.


Note: This means that the time to get to work is 1 hour and 10 minutes (or 70 minutes)


d=rt Go back to the distance rate time formula


56=r%281%29 Plug in d=56 and t=1 (which is the time coming home)


56=r Multiply



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Answer:


So the solution is r=56


This means that the average speed coming home is 56 miles per hour.